In search of H-theorem for Ulam’s redistribution of energy problem

Abstract

We discuss the possibility of deriving an H-theorem for the nonlinear discrete time evolution known as Ulam’s redistribution of energy problem. In this model particles are paired at random and then their total energy is redistributed between them according to some probability law. It appears useful to represent the evolution as a combination of two processes. The first is a linear transformation of two-particle distribution function due to redistribution while the second one is a kind of “reduction” which corresponds to new random pairing. Then information theory approach leads to a general inequality for the Ulam’s problem, which may be viewed as a kind of Clausius inequality. However, only for a special set of redistribution laws, given by symmetric beta distributions, this inequality results in the H-theorem. The H-functional in this case differs from the usual entropy by an additional term that vanishes only for the uniform redistribution law.